1. Wind Turbine Models for Solving
Electrical Power System Problems
Environmental Modeling of power system with
wind power generation
ICGST
Mohamed Ahmed
PhD Candidate
University of Waterloo
m2sadek@uwaterloo.ca

2. Part 1:
Distribution Systems

3. Distribution System Problems
1) Load Flow (Voltage Profile and Power Losses)
a) Optimum allocation of DGs or WT-DGs
b) Optimum allocation of Voltage Regulators
2) Reliability Indices (SAIFI, SAIDI, and EENS)
3) Protection schemes modifications
4) Upgrading system cost

4. Weibull Distribution














−





=
− rr
ccc
r
f
ωω
ω exp)(
1
r (shape index) =
086.1−






meanω
σ
c (scale index) =
)
1
1(
r
mean
+Γ
ω














−−=
r
c
F
ω
ω exp1)(

5. Power – Speed relationship







≥
≤≤
≤≤
≤≤
=
−
−
−
−
outcut
outcutratedrated
ratedincut
incut
w
P
h
P
ωω
ωωω
ωωωω
ωω
ω
0
)(
00
)(
ratedPaaah )()( 2
321 ωωω ++=

6. Wind Turbine Probability Model
By applying the fundamental theorem of calculating the probability distribution
of a function of a random variable, the probability p(Pw) of the active power
produced by WTs can be determined when the probability distribution of the
wind speed is known. It follows that the WT production falls into one of the
next categories:
a)For Pw=0, when
p(Pw=0)=p(ω<ωcut-in)+p(ω>ωcut-out) = F(ωcut-in)+(1-F(ωcut-out))
b) For Pw=Prated, when
p(Pw=Prated)=p(ωrated<ω<ωcut-out) = F(ωcut-out)-F(ωcut-in)
outcutincut and −− ≥≤≤ ωωωω0
outcutrated −≤≤ ωωω

7. Probability Model Cont’d
c) For Pw=h(ω), when
For wind speed variations in a short interval ωn-1<ω<ωn the
corresponding active power produced by WT ranges between,
respectively. The probability of generating Pw, which is the
average of Pn-1 and Pn is p(Pw) and can be approximated as:
ratedincut ωωω ≤≤−
)()(
)()
2
(
1
1
1
−
−
−
−=
≤≤=
+
=
nn
nn
nn
w
FF
p
PP
Pp
ωω
ωωω

8. System Under Study
Although the selection of location of WT
generators is entirely arbitrary in my work,
there is no loss of generality. In a real-life
system, the WT generators would be
located at a bus after carrying out detailed
planning studies and techno-economic
validation, which is beyond the scope of my
research.
The WT is considered to be located at one
of the remote buses (bus-18, 22, 25 or 33).
Three different WT models are considered.

9. Wind Turbine Models
Model-1: Constant power factor model
The commonly used WT-DG model is the constant power factor model.
PW and pfWT are the specified real power output and power factor for WT
installed at bus i and Qw is the calculated reactive power output. Using the
N values of active power Pw , the N values of reactive power Qw can be
obtained.
)(tan(cos)()( 1
WTww pfPQ −
×= ωω

10. Wind Turbine Models Cont’d
Model-2: Variable reactive power model
By using induction generator based WT-DGs and knowing the active power
Pw the reactive power consumed by a WT can be represented as a function of
its real power as:
Where V is the bus voltage, Pw is the real power (positive when injected into
the grid), X is the sum of the stator and rotor leakage reactances, Xc is the
reactance of the capacitors bank while Xm is reactance of the induction
generator.
)(
)(
))(()( 2
2
2
ωωω w
i
i
mc
mc
w P
V
X
V
XX
XX
Q +
−
=

11. Wind Turbine Models Cont’d
Model-3: Constant voltage model
This model is used for large scale controllable WT-DGs, the specified
values for this model is Pw and bus voltage magnitude. A capacity cap
is imposed on the WT rated apparent power.
)()()( 22
ωωω www SQP ≤+

12. Probabilistic Distribution Load Flow (PDLF)
(For PQ Models)
Backward sweep: the nodal current injected in iteration k, at node i, , is
At iteration k, starting from the branches that are connected to end nodes
and moving towards the branches connected to the substation node the
current in a branch L, IL is calculated as:
niVY
V
S
I k
iik
i
ik
i ,...,2,1)(
)(
)( )1(
)1(
)(
=∀−





= −
∗
−
ω
ω
ω
∑∈
−=∀+−=
Lm
m
k
i
k
L bbLIII 1,....,1,)()()( )()(
ωωω

13. PDLF Cont’d
Forward sweep: Nodal voltages are updated in a forward sweep from
branches connected to substation node toward the end nodes. For each
branch L, the voltage at node i is calculated using the updated voltage at
the previous node and branch currents are calculated in preceding
backward sweep as follows:
ZL is the series impedance of branch L. The previous steps are repeated
until convergence is achieved. Then the expected voltage Vi can be
calculated using the following equation:
bLIZVV k
LL
k
h
k
i ,...,2,1)()()( )()()(
=∀−= ωωω
∑=
×=
N
s
iwi VPpV
1
)())(( ωω

14. Probabilistic Compensation-Based Power Flow
(for PV Model)
The basic idea of this method can be summarized in the following steps:
1) Construct PV node sensitivity matrix ZV. In ZV, the diagonal elements are
the sum of the impedances of lines which can be formed from the PV node to
the feeder node and the off-diagonal elements are the sum of the impedances
in the lines connecting two PV nodes.
2) Perform backward current and forward voltage sweep iterations as
discussed in PDLF. If the maximum power mismatch for all buses is less than
the power convergence criterion, then proceed to step 3.
At the any iteration k, the reactive power injection required by the WT at the
bus i (Qw) to maintain its voltage at a specific value can be calculated using:
[ ] 11
21
21
)( −−
−−
−−
+−
−
−
= k
i
k
iik
i
k
i
k
i
k
i
w QVV
VV
QQ
Q specified
ω

15. Probabilistic Compensation-Based Power Flow
(for PV Model)
3) Calculate PV node voltage mismatch ΔVi. For PV node i
, the magnitude of specified voltage at node i, and Vi (ω) is the voltage at
the PV node of the final iteration of step 2. If the maximum PV node voltage
mismatch is greater than the PV node voltage convergence criterion ε, update
PV node current injection Iq,i(ω) using the following equations and then go to step
2, otherwise, the final power flow has been obtained.
specifiediii VVV −=∆ )()( ωω
specifiediV
)(][)( 1
, ωω iViq VZI ∆=∆ −
))(
2
(
,, )()(
ωδ
π
ωω ivj
iqiq eII
+
×∆=
Iq,i(ω) is to be added to or subtracted from the load current at bus i and this
is based on the sign of ΔVi(ω). If ΔVi(ω) is positive, less reactive power
generation will be injected into the PV node while if ΔVi(ω) is negative, more
reactive power generation will be injected into the PV node.

16. Sample Results
Model – 1 (WT at 18)
DLF PDLF Actual
V18 0.912 0.920 0.919
V22 0.983 0.990 0.988
V25 0.962 0.968 0.969
V33 0.908 0.916 0.914
Power loss 0.042 0.016 0.019
Substation Power, P 0.462 0.368 0.372
Substation reactive power, Q 0.312 0.240 0.251

17. Sample Results

18. Conclusions
• A probabilistic tool has been developed to solve the load flow
problem for a distribution system to examine the effects of wind
generation penetration.
• The proposed algorithm is applied to steady state analysis of a
realistic distribution feeder with dispersed wind power generation in
order to assess the effects of wind turbine operation on distribution
power loss and voltage profiles.
• Test results show that the proposed method can be effectively and
efficiently used to analyze the penetration of WTs to distribution
feeders.

19. Part 2:
Power Systems

20. Unit Commitment Problem
The Unit Commitment is basically an optimization problem to minimize
the system operation cost while satisfying system and generation
constraints.
When you solve this optimization problem, you come up with the
answer of the following questions:
1)Which ?
2)When ?
3)How much ?

21. Wind Data Clustering and Classifications
Historical, hourly, wind speed data (ωk,d,m,y) is clustered to develop a
wind speed profile for each month. A monthly wind speed profile is
obtained by averaging the wind speed for each hour of the month
(ϖk,m), over the entire data set of T years (8). Subsequently, the hourly
standard deviation of the monthly wind speed profile (σk,m) is obtained
using (9).
Where, the shape index r the scale index c of the clustered wind speed
PDF for each hour, of a given month, is obtained. The corresponding
CDF for the clustered wind speed PDF is obtained.
TDm
T
y
Dm
d
ymdk
mk
×
∑ ∑
=
= =1 1
,,,
,
ω
ϖ
2
,
1 1
,,,, )(
1
mk
T
y
Dm
d
ymdk
m
mk
TD
ϖωσ −∑ ∑
×
=
= =

22. Wind Data Clustering and Classifications Cont’d

23. Monte-Carlo Simulation
The Monte-Carlo Simulation (MCS) method requires a sequential string of
wind speed data which can be generated either from historical data or
synthetically.
The CDF of the clustered wind speed profile, F(ϖ), and hence the function [1-
F(ϖ)]will lie in the range [0,1]. By considering F(ϖ) to be a uniformly
distributed random variable in [0,1], a (MCS) can be carried out to arrive at a
large set of wind speed values for each hour, ωk,i,n, ∀ n ∈ N.
This procedure is repeated to generate several scenarios for each WF in
order to reach the desired accuracy. The MCS stops, after N simulated
scenarios, when the ratio of standard deviation of the sample mean of wind
speed at given hour of interest to the sample mean of the same hourly wind
speed becomes less than certain predetermined tolerance (ε).

24. Scenario-Reduction Using Forward Selection
Method
The Forward Selection Algorithm works recursively, until the preserved
number of scenarios S is selected. Let Ψs (s = 1, 2… N) denote the N
scenarios, such that,
Each scenario Ψs has an equal probability λs of 1/N, and Δs,s’ is the
distance of a scenario pair (s, s’), defined as follows:
),,(
..........
),,,(
),,,(
),,,(
3000,102000,101000,10000,10
3323133
3222122
3121111
WFWFWF
WFWFWF
WFWFWF
WFWFWF
PPP
PPP
PPP
PPP
−−−
−−−
−−−
−−−
=Ψ
=Ψ
=Ψ
=Ψ
Wsisiss IiPP ∈∀−=∆ ',,',

25. Scenario-Reduction Using Forward Selection
Method Cont’d
The Forward Selection Algorithm is described as follows:
1) Define set Г, such that Г = {Ψ1, Ψ2, …,ΨN}
2) Let Ω be the set of scenarios to be deleted. Set Ω is a null set at the
outset.
3) Compute distances of all scenario pairs Δs,s’ where s, s’ ∈ 1,…,N.
4) Compute Φk = Σu≠k λuΔu,k where u, k = 1,…,N
5) Identify u = η, such that, Δu,k is minimum.
6) Identify k= ξ, such that Φk is minimum.
7) Update set Г to exclude the scenario corresponding to s = ξ and hence, Г
= {Г - Ψξ}.
8) Update set Ω to include the scenario corresponding to s = ξ and hence,
Ω = {Ω + Ψξ}.
9) Update λη to be λη + λξ.
Repeat Steps 3 to 7 till the preserved number of scenarios S is attained.

26. System Under Study
The high-voltage transmission
system is represented as a three-
area system, where each area is
modeled using the IEEE RTS-32
system. The three areas are
interconnected by tie-lines.There are
three wind farms denoted by WF-1,
WF-2 and WF-3, and they are
injecting power directly to the
transmission system, at specific
buses 101, 201 and 301 respectively.
Each WF is considered to comprise
10 wind turbines (WTs), of type
VESTAS V82-1.65 MW, with a total
capacity of 16.5 MW.

27. Objective Function
∑ ∑ ∑∈ ∈ =
∈ 







×++=
Kk Jj
M
kjkj
d
kj
u
kjSs
j
stepccC
1
,,,,,,

µ

28. Stochastic LMP Model - Model Constraints
• Demand Supply Balance
• Transmission Constraint
• Spinning Reserve Constraint
• Generation Limits
• Hydro-Generation Constraint
• Ramp-up and Start-up Ramp
• Ramp-down and Shut-down Ramp Rate Constraint
• Minimum Up-time and Down-time Constraints

29. Effect of Large Capacity WFs
In this case study, the level of total wind generation capacity is increased
to 20% of total generation capacity of the system. Each WF is now
considered to comprise 100 WTs, of type VESTAS V82-1.65 MW, with a
total capacity of 165 MW. The WTs are assumed to be placed in a
rectangular configuration, arranged in 5 rows with each row having 20
WTs. The WF capacity has a significant effect on social welfare and Fig.9
presents a comparison of a typical WF (10 WT) with a large WF (100 WT).

30. Wake-Effect
( )




















+
−−
−×=
20
21
11
1
D
x
w
CThrust
x ωω
The wake effect is taken into consideration by assuming that the wind speed
profile is identical for WTs sharing the same row, while the wind profile differs
across rows. This results in a different operating condition for each row.
Therefore, a given WF can be considered to be an aggregate of five
equivalent WTs, each of 33 MW (20 x 1.65 MW) capacity. The wake effect is
modeled by reducing the incident wind speed values from one row to the next
row, in the direction of the incident wind using:

31. Environmental Constraints
Furthermore, the effect of wind generation on reduction of
environmental emissions from the system has been examined, both in
the case of the UMP and the LMP markets. The effect on market prices
will be also examined for both cases when emission caps are imposed
by external agencies, on the ISO.
Em.
Gen.
SOx NOx PM10 CO VOC Lead CH4 N2O CO2
G1 0.2 0.5 0.036 0.11 0.04 0.000014 0.002 0.004 160
G3 0.019182 0.430435 0.516522 0.02 0.003 0.000507 0.001 0.004 210
G9 0.016807 0.5 0.1 0.04 0.007 9.86E-06 0.002 0.004 170
G22 1E-08 1E-08 1E-08 1E-08 1E-08 1E-08 1E-08 1E-08 1E-08

32. Sample Results
Percentage of
wind capacity
Total expected cost variations

33. Thank You….

34. Questions ?